Answer to Question #128970 in Real Analysis for Neha Rasaily

Question #128970

the following statements true or false? Give reasons for your answer.

 a) For the function f, defined by f(x) =4x2-4x2- 7x -2,there exists a point 

C ∈ ]-1/2,2[ satisfying f′(c) = 0

 b) For all even integral values of n,


lim (x+1)-n exists.

x→∞

 c) The function f defined by f(x)= [x − 1], (where [x] is the greatest integer 

function) is integrable on the interval [2,-4].

 d) Every infinite set is an open set. 

 e) All strictly monotonically decreasing sequences are convergent.


1
Expert's answer
2020-08-12T18:28:56-0400

a). "f'(c)=-7" for all "c\\in{\\mathbb{R}}" . Thus, the statement is false.

b). We assume that "n>0" . For "n=2,4,..." we have

"\\lim_{x\\rightarrow\\infty}\\frac{1}{(1+x)^n}=0."

For "n=-2,-4,..." "\\lim_{x\\rightarrow\\infty}\\frac{1}{(1+x)^n}=+\\infty."

c). The function is integrable, since it has a finite number of discontinuity points. Namely, "f(x)" has discontinuities at n=-3,-2,-1,0,1.

d). The statement is wrong. E.g., we can take integer numbers and a ball of an arbitrary radius with a center at the point 1. It will never belong to the set of integers irrespectively of the radius.

e). We can take the sequence "-n" , where n=1,2,... It decreases and is not convergent.


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